Chapter 4.6, Problem 13P

Furnish the details in determining when the gain function given by Eq. $\left(10\right)$ is maximum, that is, show that $w_{\max }^2$ and $\left|G\left(i{w}_{\max }\right)\right|$ are given by Eqs. $\left(14\right)$ and $\left(15\right)$, respectively.

$G\left(iw\right)=\left|G\left(iw\right)\right|e^{-i\varphi \left(w\right)}=\left|G\left(iw\right)\right|\left(\cos \left(\varphi \left(w\right)\right)-i\sin \left(\varphi \left(w\right)\right)\right)$,          $\left(10\right)$

$w_{\max }^2=w_0^2-2{\delta }^2=w_0^2-\frac{{\gamma }^2}{2m^2}=w_0^2\left(1-\frac{r^2}{2mk}\right)$,      $\left(14\right)$

$\left|G\left(i{w}_{\max }\right)\right|=\frac{m}{\gamma w_0\sqrt{1-\left({\gamma }^2/4mk\right)}}\approx \frac{m}{\gamma w_0}\left(1+\frac{{\gamma }^2}{8mk}\right)$,          $\left(15\right)$